@Lu_sg these mean the same. Frantic is more intense. Frantic means that they were very anxious and afraid. Perhaps even to say they were not thinking clearly. Desperately means they really wanted to find the child.
So frantic implies “they” refers to parents. “Desperately” implies “they” were friends of police.
“The parents were frantically searching. The police and friends searched Desperately.”
@Lu_sg I will explain a different way. Frantic means very desperate. So this is a question of degree. So this is same difference as “ he is starving” and “he is hungry.” Starving is much stronger word.